Elliptic curves over \(\Q(\sqrt{5}) \) of conductor 784.1

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Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
784.1-a1	784.1-a	$1$	$1$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	784.1	\( 2^{4} \cdot 7^{2} \)	\( 2^{16} \cdot 7^{2} \)	$1.05731$	$(2), (7)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓						$1$	\( 1 \)	$1$	$2.007136303$	0.897618643	\( -\frac{221184}{7} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -8 \phi - 8\) , \( -16 \phi - 12\bigr] \)	${y}^2={x}^{3}+\left(-8\phi-8\right){x}-16\phi-12$
784.1-b1	784.1-b	$4$	$4$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	784.1	\( 2^{4} \cdot 7^{2} \)	\( 2^{16} \cdot 7^{8} \)	$1.05731$	$(2), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2 \)	$1$	$6.351380586$	1.420211874	\( -\frac{574992}{2401} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11 \phi - 11\) , \( 72 \phi + 54\bigr] \)	${y}^2={x}^{3}+\left(-11\phi-11\right){x}+72\phi+54$
784.1-b2	784.1-b	$4$	$4$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	784.1	\( 2^{4} \cdot 7^{2} \)	\( 2^{16} \cdot 7^{2} \)	$1.05731$	$(2), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 1 \)	$1$	$12.70276117$	1.420211874	\( -\frac{1042385925888}{7} a + \frac{1686615870096}{7} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 19 \phi - 86\) , \( -152 \phi + 278\bigr] \)	${y}^2={x}^{3}+\left(19\phi-86\right){x}-152\phi+278$
784.1-b3	784.1-b	$4$	$4$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	784.1	\( 2^{4} \cdot 7^{2} \)	\( 2^{8} \cdot 7^{4} \)	$1.05731$	$(2), (7)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	\( 2 \)	$1$	$25.40552234$	1.420211874	\( \frac{28311552}{49} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -16 \phi - 16\) , \( 44 \phi + 33\bigr] \)	${y}^2={x}^{3}+\left(-16\phi-16\right){x}+44\phi+33$
784.1-b4	784.1-b	$4$	$4$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	784.1	\( 2^{4} \cdot 7^{2} \)	\( 2^{16} \cdot 7^{2} \)	$1.05731$	$(2), (7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 1 \)	$1$	$12.70276117$	1.420211874	\( \frac{1042385925888}{7} a + \frac{644229944208}{7} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -19 \phi - 67\) , \( 152 \phi + 126\bigr] \)	${y}^2={x}^{3}+\left(-19\phi-67\right){x}+152\phi+126$

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  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.